>Finance

Created 7/1/1996
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Business Administration 130

Brealey and Myers, chapter 8: Introduction to Portfolio Theory

http://www.j-bradford-delong.net/Intro_Finance/BAonethirty8.html

Basics:

• Present value of a perpetuity: C/r
• Present value of a growing (or shrinking) perpetuity: C/(r-g)
• Present value of C dollars t years from now: C/[(1+r)^t]
• Present value of a C-dollar t-year annuity: C[(1/r)-(1/[r(1+r)^t])
• beta = [E((r₁-E(r₁))(m-E(m))]/E[(m-E(m))²]
• r*_i = r*_f + beta_i(r*_m-r_f)

Not happy with the lecture I gave last time; tried to do to much and gave much-too-quick explanations of important things. So let me back up a bit, and go over the "benefits of diversification" again.

Benefits of Diversification

"The opportunity cost of capital depends on the risk of the project": I've been saying this for three and a half weeks now. But what does it mean? What is the risk of a project? Why should the appropriate cost of capital vary depending on how risky the project is?

Let's start with risk.

State of the World	Mega Manufacturing	Startup Semiconductor
HH	+40%	+10%
HT	+10%	-20%
TH	+10%	+40%
TT	-20%	+10%
Expected Value...	EV = +10%	EV = +10%
Standard Deviation...	SD = 21.2%	SD = 21.2%

What risk-return combination do you get if you put all your money into one stock or the other?

But suppose you start mixing one with the other...

25%M+75%S	Mega Manufacturing	Startup Semiconductor
17.5%	+40%	+10%
-12.5%	+10%	-20%
32.5%	+10%	+40%
2.5%	-20%	+10%
EV = +10%	EV = +10%	EV = +10%
SD= 16.8%	SD = 21.2%	SD = 21.2%

	0M+1S	.25M+.75S	.5M+.5S	.75M+.25S	1M+0S
Expected Return	+10%	+10%	+10%	+10%	+10%
Standard Deviation	21.2%	16.8%	15%	16.8%	21.2%

More General Formulas for Variances:

Unpack this formula: what does it mean?

An especially interesting special case:

Suppose we have a number of securities, indexed by i, all of whose returns look something like this:

and suppose that the u_i's are uncorrelated with each other and with the market--that their covariance is zero, or is at least not too large.

Now suppose that we take our portfolio, and our portfolio shares x_i, and set each x_i=1/N: put 1/N of our portfolio's wealth into each of the first N of these securities.

What does our variance look like?

Well, we know that:

And we can substitute in:

Noticing that the first and third terms look awfully familiar:

is the variance of the portfolio:

• Now the first term is simply the average beta of the stocks in the portfolio, squared, times the variance of the "market" portfolio...
• The second term is the sum of the idiosyncratic risks u associated with the securities i--all divided by N-squared.

Now a funny thing happens as N gets large: the second term vanishes: the sigmas stay (roughly) the same size, and you are adding up more (N) of them, but each one is divided by N-squared. So as N approaches infinity, the second term gets small--eventually small enough to ignore...

An even funnier thing happens if we consider the variance of a portfolio made up of a large number of stocks (N large), for which the average beta happens to be one.

• Then the first term is simply sigma-squared-m, and the second term is negligible: the variance of the portfolio is the variance of the "market"

Conclusion:

If an investor is doing his or her job--trying to get to a portfolio that has the minimum risk for a given expected return--then "idiosyncratic" risk can be diversified away: by putting all your eggs into many baskets, you essentially eliminate any idiosyncratic risk factors from your portfolio.

Hence a security's "riskiness"--from the point of view of its impact on the overall riskiness of your portfolio as a whole--is summarized by its beta...

Risk and Return

Now let me move on to risk and return proper...

"State of the World"	Universal Utility	Mega Manufacturing	Exciting Exports	Startup Semiconductor
HH	+20%	+40%	+40%	+20%
HT	+0%	+10%	+50%	-20%
TH	+10%	+10%	-30%	-20%
TT	-10%	-20%	-20%	+20%

Suppose I am choosing portfolios from Mega Manufacturing and Startup Semiconductor:

I get the highest return from Mega Manufacturing, but I can get a lower standard deviation--at not much cost in return--by starting to diversify.

Suppose I diversify among the four different stocks:

I can construct a large number of truly, truly putrid portfolios (in fact, by throwing money into the sea I can construct even worse portfolios...)

I can also get outside the frontier of the parallelogram defined by the risk/return characteristics of the individual stocks...

Introducing lending and borrowing:

Choose the portfolio S that just touches the line that goes through the riskfree-rate point and lies entirely to one side of the feasible portfolio set.

Then borrow (and lend) until you get to the risk-return characteristics you want...

Capital asset pricing model...